A324539 -id:A324539 - cp's OEIS frontend

A327153 Number of divisors d of n such that sigma(d)*d is equal to n.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

a(n) tells how many times in total n occurs in A064987.

			336 has 20 divisors: [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336]. Only two of them, d=12 and d=14, satisfy sigma(d) = (336/d), thus a(336) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A064987, A327165 (positions of nonzero terms).

Cf. also A324539.

A327153(n) = sumdiv(n, d, (n==d*sigma(d)));

a(n) = Sum_{d|n} [A000203(d)*d == n], where [ ] is the Iverson bracket.

A355944 a(n) = smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 4, 5, 2, 7, 8, 3, 5, 11, 4, 13, 7, 5, 16, 17, 3, 19, 8, 14, 11, 23, 8, 10, 13, 9, 28, 29, 5, 31, 32, 11, 17, 7, 4, 37, 19, 26, 8, 41, 14, 43, 44, 5, 23, 47, 16, 35, 10, 17, 52, 53, 9, 11, 32, 38, 29, 59, 8, 61, 31, 21, 64, 13, 11, 67, 68, 23, 7, 71, 16, 73, 37, 10, 76, 33, 26, 79, 16, 27, 41, 83, 28, 17, 43

Offset: 1

Antti Karttunen, Jul 27 2022

a(n) is the smallest positive k such that A324580(k) is a multiple of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A276086, A324539, A324540, A324541, A324580, A355945, A356151, A356152, A356153, A356160 (fixed points, where a(n)=n), A356161.

Cf. also A344005, A356164.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355944(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(k)));

a(n) = n - A355945(n).

A324540 Numbers not in range of A324580.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

Positions of zeros in A324539.

Cf. A324541 (complement).

Cf. A276086, A324539, A324580, A065091 (a subsequence).

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324539(n) = sumdiv(n,d,(d==A276086(n/d)));
for(n=1,100,if(0==A324539(n), print1(n, ", ")));

search_limit = 15000;
A324580(n) = n*A276086(n);
A324540list(search_up_to) = { my(v=vector(search_up_to),c=0,k); for(n=1,#v,k=A324580(n); if(k<=#v && !v[k], v[k] = n; c++)); my(u=vector(#v-c), j=0); for(n=1,#v,if(0==v[n], j++; u[j] = n)); (u); };
v324540 = A324540list(search_limit);
A324540(n) = v324540[n];

A324541 Numbers that occur in range of A324580.

Original entry on oeis.org

0, 2, 6, 18, 30, 36, 70, 90, 120, 210, 270, 300, 434, 450, 650, 672, 990, 1050, 1260, 1386, 2142, 2250, 2310, 2590, 2940, 3600, 3990, 4410, 4642, 4750, 5978, 6996, 7350, 7500, 7650, 8190, 9114, 11880, 12600, 14058, 15000, 15050, 15750, 16170, 18480, 18522, 21186, 23100, 23870, 24750, 25830, 28224, 30030, 30870, 31250, 32830, 35970, 37114, 42000

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Indexing begins from 0 because the term a(0) = 0 is a special case.
Sequence A324580 sorted into ascending order, with duplicate occurrences removed. The first such duplicate is 2250 = A324580(15) = 150*15 = A324580(18) = 125*18. The next is 5402250 = A324580(105) = A276086(105)*105 = A324580(125) = A276086(125)*125.
Terms after zero are the positions of nonzero terms in A324539.

Cf. A324540 (complement).

Cf. A002110 (a subsequence), A276086, A324539, A324579, A324580.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324539(n) = sumdiv(n,d,(d==A276086(n/d)));
for(n=1,oo,if(A324539(n)>0, print1(n, ", "))); \\ Print terms after zero.

\\ This program is better for computing many terms:
search_limit = 9699690;
A324580(n) = n*A276086(n);
A324541list(lim) = { my(s=Set([]),k); for(n=1,lim, k=A324580(n); if(k<=lim, s = setunion([k], s))); Vec(s); };
v324541 = A324541list(search_limit);
A324541(n) = if(!n,n,v324541[n]);

cp's OEIS Frontend

A327153 Number of divisors d of n such that sigma(d)*d is equal to n.

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A355944 a(n) = smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

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A324540 Numbers not in range of A324580.

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A324541 Numbers that occur in range of A324580.

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